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Considering firm’s innovation input of green products and channel service, this paper, in dynamic environment, studies a dynamic price game model in a dual-channel green supply chain and focuses on the effect of parameter changing on the pricing strategies and complexity of the dynamic system. Using dynamic theory, the complex behaviors of the dynamic system are discussed; besides, the parameter adaptation method is adopted to restrain the chaos phenomenon. The conclusions are as follows: the stable scope of the green supply chain system enlarges with decision makers’ risk-aversion level increasing and decreases with service value increasing; excessive adjustment of price parameters will make the green supply chain system fall into chaos with a large entropy value; the attraction domain of initial prices shrinks with price adjustment speed increasing and enlarges with the channel service values raising. As the dynamic game model system is in a chaotic state, the profit of the manufacturer will be damaged, while the efficiency of the retailer will be improved. The system would be kept at a stable state and casts off chaos by the parameter adaptation method. Results are significant for the manager to make reasonable price decision.

At present, China is the largest producer and consumer of household appliances in the world. The development of the home appliance industry has made great contribution to the economic development. According to the relevant report of China’s home appliance market in 2019, the scale of China’s home appliance market still keeps rapid growth [

In recent years, many scholars have conducted extensive research on technological innovation for green product in various ways [

The previous literature has elaborated the influence of innovation factors on the supply chain from various angles in detail, and the conclusions have important reference value for society and enterprises. However, the complexity degree of the supply chain based on the impact of product innovation input on market demand is rarely explored.

In recent years, the supply chain-related problems considering channel service factors have become the focus of the industry and academia [

Scholars have done some research on price game issues. Xin and Sun [

This paper establishes a price game model considering the factors of green innovation input and channel service in decentralized decision scenario. Using game theory and nonlinear dynamics theory, we discuss the equilibrium points and the complex dynamic behaviors of the dynamic system and study the effects of channel service level and risk-aversion level on optimal pricing, stability, and utility of the dual-channel value chain system. The global stability and chaos phenomenon of the dynamic system are analyzed.

Our theoretical contribution is as follows: the first contribution is to the construct dynamic dual-channel supply chain model considering innovation investment and channel service. The second contribution is to analyze the effects of the key parameters on the stability and profitability of the dynamic dual-channel supply chain model.

The rest of this paper is arranged as follows: in Section

In a dual-channel green supply chain, to satisfy consumers’ demand for intelligent and personalized green products, the manufacturer makes innovative investment in green products, and

The dual-channel green supply chain system.

To support this research, the following assumptions are made in this paper:

We assume that the participants all show risk-aversion behaviors facing the changing market demand.

In a game cycle, because of difficulty that decision makers grasp the perfect market information through their own abilities, therefore, this paper assumes that the manufacturer and retailer are both bounded rationality and they will make the next decision based on the current marginal utility; as the current marginal utility is more than zero, they would improve selling price in the next period; otherwise, they would reduce selling price.

For the convenience of research, this paper assumes that the unit distribution cost of participants is zero [

The notations of parameters and its meanings employed in this paper are listed in Table

Notations and their meanings.

The basic market scale | |

The mean of basic market scale | |

Consumer’s preference for direct channel | |

The wholesale price of product | |

Manufacturing cost per unit product | |

Retail price of product in direct channel | |

Retail price of product in traditional channel | |

Price-sensitive coefficient | |

Cross-price-sensitive coefficient | |

The scale of innovation input | |

Service value of unit product of the manufacturer | |

Service value of unit product of the retailer | |

Service cost coefficient of the manufacturer | |

Service cost coefficient of the retailer | |

Service cost of unit product for manufacturer | |

Service cost of unit product for the retailer | |

Market demand variance | |

Customer demand created by innovation input | |

Innovation input coefficient | |

Risk attitude of the manufacturer | |

Risk attitude of the retailer |

In the market competition, market demand is not only affected by retail price but also by the manufacturer’ investment in technological innovation and channel services provided by the participant. In this paper, considering the above factors and relevant literature [

The connection between service value and service cost of unit product satisfies

Therefore, service cost and service value change in the same direction.

Based on the features of this paper, the parameters should satisfy

According to inequality (

The profit functions of firms are as follows:

In the face of the uncertain demand, the manufacturer and the retailer have financial risks for their product sales [

The expected utilities of participants are as follows:

Making first-order derivatives of

Making second-order derivatives of

From the above analysis, the manufacturer’s utility function

From (

Substituting (

Considering the first-order partial derivatives of

In multistage game process, participants cannot grasp the perfect market information of the current market, and the grey forecasting model is a good way for the nonlinear system [

For the system disturbed by weak noise, its variation can be seen as the random factors around the deterministic system and transformation between them [

Based on the theory of the fixed point [

Nash equilibrium point

See Appendix.

Nash equilibrium point

See Appendix.

Economically, zero price means nothing to manufacturers and retailers. Next, we will study the stability characteristics of equilibrium solutions (

The Jacobian matrix of dynamic system (

The feature equation of the Jacobian matrix can be written as follows:

In (

(see [

If all eigenvalues of

Next, we adopt numerical simulation to indicate how key factors affect the complex dynamical behaviors of dynamic system (

In dynamic system (

Figure

The stable region of dynamic system (

Figure

The stable regions of dynamic system (

Figure

For understanding better the change process of dynamic system (

The 2D bifurcation diagram of dynamic system (

Figures

The 2D bifurcation diagrams in the (

The 2D bifurcation diagrams in the (

From the above analysis, a conclusion can be drawn that the stability region of dynamic system (

If participants improve their service levels in order to obtain the best utilities, they will consume a lot of manpower and material resources of the dual-channel green supply chain. At this time, the ability of the dual-channel green supply chain system to resist risks will be weakened, which will lead to the decrease of system stability and the increase of vulnerability. Therefore, from the point of view of the supply chain, decision makers should make their own service decisions and price adjustment decisions prudently to make the dynamic system stable.

The variation of the key parameter affects the stability of system (

With the same values for parameters, the basins of attraction about initial price

Basins of attraction of dynamic system (

Basins of attraction with

Since the influence of the price adjustment parameters of participants on system behavior is similar, the evolution of system behavior is discussed by taking the price adjustment parameter of the manufacturer as an example. Figure

The price bifurcation and entropy with

Figure

In order to visually show the effect of service values on the behavior of the system, Figures

The bifurcation diagrams of dynamic system (

Figure

The chaotic attractor of dynamic system (

Figure

Sensitivity to the initial value is also an important characteristic of chaotic systems. Figure

Sensitivity of dynamic system (

From this, it can be drawn that the sensitivity of the dynamic system in the unstable state is just like the butterfly effect. With the small change of initial conditions, the chaotic system will fluctuate violently with time. In formulating market strategies, decision makers should choose the initial values carefully.

Thus, the excessive price adjustment parameter and larger service value can easily make dynamic system (

Figure

The expected utility of dynamic system (

The evolution of expected utility over time in different periods. (a)

In Figure

The average utility diagram with

The above research states clearly that the average utility of the manufacturer is higher than that in the stable state, and the one of the retailer is lower than that in the unstable state. In this uncertain situation, it brings great suffering to the participants in making the next price decision. If the chaos phenomenon is not controlled, it will lead to a vicious circle in the market and even lead to the withdrawal of participants from the market competition. In order to restore market stability, it is necessary to control the chaotic market effectively.

Some chaos control methods have been applied to the supply chain, such as modified straight-line stabilization method [

Figures

The bifurcation diagram of controlled system (

When

The price bifurcation diagram and entropy of controlled system (

In market competition, chaos has an important effect on the utility of participants. However, due to the market complexity and the difference of decision makers, the behavior of the decision makers may make the stable market into a bifurcation state or even chaotic state in pursuit of maximizing their utilities. At this time, it is necessary for participants to cooperate and coordinate to hold the system in a stable state, or for the government to control the chaotic market and create a good market competition environment.

Considering the innovation input for green products and channel service, this article establishes a dynamic game model under the optimal innovation input and focuses on the influence of key factors on the pricing decisions and complexity of the dynamic system. Firstly, equilibrium points, conditions for existence, and local stability of the dynamic system are discussed. Secondly, the complexity dynamics of the dynamic system (the influence of parameters on prices, utilities, and global stability of the dynamic system) are studied by employing dynamic theory. Finally, the parameter adaptation method is employed to restrain chaos of the system. The results are summarized as follows:

The stable range of the dynamic dual-channel green supply chain system enlarges with the increase of risk-aversion levels of participants, shrinks with service value increasing. In addition, the retailer’s channel service will only decrease the stable scope of its own channel price adjustment speed and brings no influence on the manufacturer’s decision range of channel price adjustment speed.

The dynamic system experiences flip bifurcation and enters into chaos as the adjustment speed increasing. In stable range, the retail prices and the utilities of participants are fixed values, and the entropy value of the dynamic dual-channel green supply chain system is low. In the chaotic state, the average utility of the manufacturer declines and that of the retailer improves. The entropy value of the dynamic dual-channel green supply chain system is high. So, chaos is beneficial for the retailer and is harmful to the manufacturer to achieve high expected utility.

The attraction domain of initial prices shrinks with price adjustment speed increasing and enlarges with the channel service values increasing. The dynamic system, from chaos, can run to a stable state again using the parameter adaptation method.

There are some shortcomings in this paper, and we can study from the following aspects in the future: (1) one can consider the nonlinearity of market demand, which is more in line with the market competition environment. (2) One can consider the cost sharing of innovation input based on the cooperation of participants.

the Jacobian matrix of dynamic system (

Substituting the values of

substituting

The eigenvalues of the matrix

According to the previous restrictions on the parameters, it can deduce that

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Li Qiuxiang provided research methods, Huang Yimin wrote the original draft, and Li Mengmeng revised the paper.

The research was supported by the National Social Science Foundation of China (no. 19FGLB067) and Henan Province Soft Science Research Plan Project (no. 192400410088).